Let w(s,t)=f(u(s,t),v(s,t)) where u(1,0)=−6,∂u∂s(1,0)=5,∂u∂1(1,0)=7 v(1,0)=−8,∂v∂s(1,0)=−8,∂v∂t(1,0)=6 ∂f∂u(−6,−8)=−1,∂f∂v(−6,−8
Blababa [14]
![w(s,t)=f(u(s,t),v(s,t))](https://tex.z-dn.net/?f=w%28s%2Ct%29%3Df%28u%28s%2Ct%29%2Cv%28s%2Ct%29%29)
From the given set of conditions, it's likely that you are asked to find the values of
![\dfrac{\partial w}{\partial s}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20w%7D%7B%5Cpartial%20s%7D)
and
![\dfrac{\partial w}{\partial t}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20w%7D%7B%5Cpartial%20t%7D)
at the point
![(s,t)=(1,0)](https://tex.z-dn.net/?f=%28s%2Ct%29%3D%281%2C0%29)
.
By the chain rule, the partial derivative with respect to
![s](https://tex.z-dn.net/?f=s)
is
![\dfrac{\partial w}{\partial s}=\dfrac{\partial f}{\partial u}\dfrac{\partial u}{\partial s}+\dfrac{\partial f}{\partial v}\dfrac{\partial v}{\partial s}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20w%7D%7B%5Cpartial%20s%7D%3D%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20u%7D%5Cdfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20s%7D%2B%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20v%7D%5Cdfrac%7B%5Cpartial%20v%7D%7B%5Cpartial%20s%7D)
and so at the point
![(1,0)](https://tex.z-dn.net/?f=%281%2C0%29)
, we have
![\dfrac{\partial w}{\partial s}\bigg|_{(s,t)=(1,0)}=\dfrac{\partial f}{\partial u}\bigg|_{(u,v)=(-6,-8)}\dfrac{\partial u}{\partial s}\bigg|_{(s,t)=(1,0)}+\dfrac{\partial f}{\partial v}\bigg|_{(u,v)=(-6,-8)}\dfrac{\partial v}{\partial s}\bigg|_{(s,t)=(1,0)}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20w%7D%7B%5Cpartial%20s%7D%5Cbigg%7C_%7B%28s%2Ct%29%3D%281%2C0%29%7D%3D%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20%0Au%7D%5Cbigg%7C_%7B%28u%2Cv%29%3D%28-6%2C-8%29%7D%5Cdfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20s%7D%5Cbigg%7C_%7B%28s%2Ct%29%3D%281%2C0%29%7D%2B%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20%0Av%7D%5Cbigg%7C_%7B%28u%2Cv%29%3D%28-6%2C-8%29%7D%5Cdfrac%7B%5Cpartial%20v%7D%7B%5Cpartial%20s%7D%5Cbigg%7C_%7B%28s%2Ct%29%3D%281%2C0%29%7D)
![\dfrac{\partial w}{\partial s}\bigg|_{(s,t)=(1,0)}=(-1)(5)+(2)(-8)=-21](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20w%7D%7B%5Cpartial%20s%7D%5Cbigg%7C_%7B%28s%2Ct%29%3D%281%2C0%29%7D%3D%28-1%29%285%29%2B%282%29%28-8%29%3D-21)
Similarly, the partial derivative with respect to
![t](https://tex.z-dn.net/?f=t)
would be found via
![\dfrac{\partial w}{\partial t}\bigg|_{(s,t)=(1,0)}=\dfrac{\partial f}{\partial u}\bigg|_{(u,v)=(-6,-8)}\dfrac{\partial u}{\partial t}\bigg|_{(s,t)=(1,0)}+\dfrac{\partial f}{\partial v}\bigg|_{(u,v)=(-6,-8)}\dfrac{\partial v}{\partial t}\bigg|_{(s,t)=(1,0)}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20w%7D%7B%5Cpartial%20t%7D%5Cbigg%7C_%7B%28s%2Ct%29%3D%281%2C0%29%7D%3D%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20%0Au%7D%5Cbigg%7C_%7B%28u%2Cv%29%3D%28-6%2C-8%29%7D%5Cdfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20t%7D%5Cbigg%7C_%7B%28s%2Ct%29%3D%281%2C0%29%7D%2B%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20%0Av%7D%5Cbigg%7C_%7B%28u%2Cv%29%3D%28-6%2C-8%29%7D%5Cdfrac%7B%5Cpartial%20v%7D%7B%5Cpartial%20t%7D%5Cbigg%7C_%7B%28s%2Ct%29%3D%281%2C0%29%7D)
1) Taking in account that the function is f(x)= -(2/3) |x+4|-6, I enclose a file with the graph.
That helps you to conclude:
a) The graph of f(x) has a vertex on (-4, -6)
b) When you multiply a function times 2/3 it is vertically compressed which is equivalent to horizontally streched.
c) The graph of f(x) opens downward
d) The domain of f(x) is all the real values (the absolute function accepts any value of x either positive or negative)
Answer: the graph of f(x) is horizontally stretched.
You can factor this by taking out the GCF (greatest common factor).
In the expression, the greatest common factor between the two terms is 5.
Take out 5:
5(2b + 5)
Is it possible to factor 2b + 5 even more?
No. So, your answer is D. 5(2b + 5).
Answer: ab =6
have:
![\frac{1}{a}+\frac{1}{b}+\frac{1}{ab}=1\\\\=>\frac{b}{ab}+\frac{a}{ab}+\frac{1}{ab}=1\\\\\frac{a+b+1}{ab}=1](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7Ba%7D%2B%5Cfrac%7B1%7D%7Bb%7D%2B%5Cfrac%7B1%7D%7Bab%7D%3D1%5C%5C%5C%5C%3D%3E%5Cfrac%7Bb%7D%7Bab%7D%2B%5Cfrac%7Ba%7D%7Bab%7D%2B%5Cfrac%7B1%7D%7Bab%7D%3D1%5C%5C%5C%5C%3C%3D%3E%5Cfrac%7Ba%2Bb%2B1%7D%7Bab%7D%3D1)
=> a + b + 1 = ab
⇔ a + b + 1 - ab = 0
⇔ b - 1 - a(b - 1) + 2 = 0
⇔ (b - 1)(1 - a) = -2
because a and b are postive integers => (b - 1) and (1 - a) also are integers
=> (b - 1) ∈ {-1; 1; 2; -2;}
(1 -a) ∈ {-1; 1; 2; -2;}
because (b -1).(1-a) = -2 => we have the table:
b - 1 -1 1 2 -2
1 - a 2 -2 -1 1
a -1 3 2 0
b 0 2 3 -1
a.b 0 6 6 0
because a and b are postive integers
=> (a;b) = (3;2) or (a;b) = (2;3)
=> ab = 6
Step-by-step explanation:
X=width of a coccer pitch
2x-20=length of a soccer pitch
Area (rectangule)=length x width
We suggest this equation:
x(2x-20)=6000
2x²-20x=6000
2x²-20x-6000=0
x²-10x-3000=0
We solve this quadratic equation:
x=[10⁺₋√(100-4*1*-3000)]/2=[10⁺₋√(100+12000)]/2=
=(10⁺₋110)/2
we have two solutions:
x₁=(10-110)/2=-50, invalid solution.
x₂=(10+110)/2=60
x=60
2x-20=2(60)-20=120-20=100
Solution: the length is 100 m, and the width is 60 m.
To check:
Area=100 m*60 m=6000 m²
The twice of width is =2(60 m)=120 m,
20 m less than twice its width is: 120 m-20 m=100 m=the length.